Related papers: Differentiating by prime numbers

We introduce signature payoffs, a family of path-dependent derivatives that are given in terms of the signature of the price path of the underlying asset. We show that these derivatives are dense in the space of continuous payoffs, a result…

The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.

A classical problem in analytic number theory is to study the distribution of $\alpha p$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is…

We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.

In this short survey we look at a few basic features of p-adic numbers, somewhat with the point of view of a classical analyst. In particular, with p-adic numbers one has arithmetic operations and a norm, just as for real or complex…

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…

We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.

We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^\theta$ for some $\theta > 0$ depending on $k$. The proof…

We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

We study the numerical differentiation formulae for functions given in grids with arbitrary number of nodes. We investigate the case of the infinite number of points in the formulae for the calculation of the first and the second…

We generalize the concept of a number derivative, and examine one particular instance of a deformed number derivative for finite field elements. We find that the derivative is linear when the deformation is a Frobenius map and go on to…

In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…

We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…

We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between…

Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…

The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime…